Averaged Models

In this section thegoverning equations determining the averaged Eulerian-Lagrangian and the Eulerian-Eulerian modeling concepts are discussed. 

1 Averaged Eulerian-Lagrangian Multi-Phase Models 

For dispersed multiphase flows a Lagrangian description of the dispersed phase are advantageous in many practical situations. In this concept the individual particles are treated as rigid spheres (i.e., neglecting particle deformation and internal flows) being so small that they can be considered as point centers of mass in space. The translational motion of the particle is governed by the Lagrangian form of Newton s second law [42, 109, 128, 158]  

The particle trajectory is calculated from the definition of the translational velocity of the center of mass of the particle  

2 Averaged Eulerian-Eulerian Multi-fluid Models 

The main advantage of the Eulerian-Lagrangian approach (i.e., compared to the alternative Euler-Euler model described in the next subsection) is its flexibility with respect to the incorporation of the microscopic transport phenomena. Particle dynamics can in principle be described in detail, a particle size distribution can easily be incorporated, direct particle-particle interactions can be accounted for as well as the hydrodynamic interaction between neighboring particles. 

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Autoregressive Integrated Moving Average Model (ARIMA) ... 

Integrated Mnvino Average Model tIMAl. Proper application of time series analysis requires that the variance of the series be constant and that there be no major trend. Any segment of the time series should be very much like any other segment. If this is not the case then the inferences will depend... 

A general approach was developed by G.E.P. Box and G.M. Jenkins (S) which combines these various methods into an analysis which permits choice of the most appropriate model, checks the forecast precision, and allows for interpretation. The Box-Jenkins analysis is an autoregressive integrated moving average model (ARIMA). This approach, as implemented in the MINITAB computer program is one used for the analyses reported here. 

A. Kurenkov and M. Oberlack 2005, Modelling turbulent premixed combustion using the level set approach for Reynolds averaged models. Flow Turbulence Combust. 74 387-407. 

Figure 72. Average modelled monthly fluid temperatures in the ground loop heat exchanger,...

It is important to point out that proposed structures are derived from structural parameters, calculated from the characterization results, so they are average models, rather than representing an identified molecule. 

Fig. 2.25 Temporal mean (left) and standard deviation (right) of the zonal mean volatilisation rate over 10 years [kg/(kg s)]. Dashed lines show volatilisation rates derived from zonal mean SST and wind speed (denoted as zonally averaging model). Solid lines show volatilisation rates derived from zonally resolved SST and wind speed, which were zonally averaged afterwards (denoted as zonally resolved model).

NRTL-SAC has been demonstrated through the case study on Cimetidine as a valuable aid to solubility data assessment and targeted solvent selection for crystallization process design. The average model error is typically 0.5 Ln (x) [1] and is sufficient as a solvent screening tool. Methods that can deliver greater accuracy would increase the value and utility of these techniques. It is impressive in the case of Cimetidine that the NRTL-SAC correlation is capable of reasonable accuracy and predictive capability on the basis of just 2 fitted parameters. Further work to extend the solvent database and optimize the descriptive parameters will be beneficial, and are planned by the developers. 

Pheromone propagation by wind depends on the release rate of the pheromone (or any other odor) and air movements (turbulent dispersion). In wind, the turbulent diffusivity overwhelms the diffusion properties of a volatile compound or mixture itself. Diffusion properties are now properties of wind structure and boundary surfaces, and preferably termed dispersion coefficients. Two models have dominated the discussion of insect pheromone propagation. These are the time-average model (Sutton, 1953) and the Gaussian plume model. 

The time-average model considers the average concentration of airborne materials at sites downwind from a point source. The concentration (or density D) of a pheromone at any one point with the coordinates x (downwind direction), j (horizontal crosswind [transverse] dimension), and z (vertical dimension) can be estimated with the following formula. 

For triggering behavior, the concentration at one point in time is more important than the average concentration. Therefore, in the real world, considerable deviation from time-averaging models is observed. In addition to timeaveraging models, peak concentrations of odors in turbulent systems have to be considered. Aylor (1976) estimated peak concentrations for air currents in forests. Average concentrations, as calculated by the Sutton formula, may be as low as only a few percent of maximmn (peak) concentrations. It is often the latter, however, that would trigger an animal s response. 

Figure 4.34 SG1524 buck regulator average model.

The phase and gain margins were measured using an average model of the SG1524 buck regulator. The schematic of the average model is shown in Fig. 4.34. 

A comparison between the step load response using the transient domain model and the state space average model is shown in Fig. 4.49, while a similar comparison of the output inductor current during the transient step load is shown in Fig. 4.50. 

Figure 4.48 Simulated tum-on of SGI524 buck regulator using average model.

Uniformist, Average Models. We divide the current water structure models into two major categories. The first treats water essentially as an unstructured liquid while the second admits the simultaneous existence of at least two states of water—i.e., the structural models which Frank has termed the mixture models. ... 

Mixture Models Broken-Down Ice Structures. Historically, the mixture models have received considerably more attention than the uniformist, average models. Somewhat arbitrarily, we divide these as follows (1) broken-down ice lattice models (i.e., ice-like structural units in equilibrium with monomers) (2) cluster models (clusters in equilibrium with monomers) (3) models based on clathrate-like cages (again in equilibrium with monomers). In each case, it is understood that at least two species of water exist—namely, a bulky species representing some... 

At first glance, the averaged model would appear to serve most researchers who are looking for a molecular model to help them explain the function of the molecule and rationalize other chemical, spectroscopic, thermodynamic, and kinetic data. On the other hand, you might think that the ensemble and distance-restraint files are of most use to those working to improve structure determination techniques. There are good reasons however, for all researchers to look carefully at the ensemble, as discussed in the next section. 

If some or all of the ensemble conformations reveal actual alternative conformations in solution, then these models contain useful information that may be lost in producing the averaged model. If the most important conformations for molecular function are represented in subsets of models within the final ensemble, then an averaged model may mislead us about function. Just like crystallographic models, NMR models do not simply tell us what we would like to know about the inner workings of molecules. Evidence from other areas of research on the molecule are necessary in interpreting what NMR models have to say. 

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